Analysis of Joint Clearances in Multibody Systems

A methodology for the study of typical smooth joint clearances in multibody systems is presented. The proposed approach takes advantage of the analytical definition of the material surfaces defining the clearance, resulting in a formulation where the gap does not play a central role, as it happens in standard contact models. The contact forces are formulated in conserving form, such that the balance of total energy during the intermittent contact is exactly established in the discrete time integration scheme. Some numerical applications are presented, showing that the proposed methodology is very stable in long-term simulations with relatively large time step sizes. Therefore, it appears to be promising in terms of efficiency and robustness for the numerical analysis of real joints with clearances.     

Modeling and simulation of the nonsmooth planar rigid multibody systems with frictional translational joints

The main purpose of this paper is to present a modeling and simulation method for the rigid multibody system with frictional translational joints. The small clearance between a slider and guide is considered. The geometric constraints of the translational joints are treated as bilateral constraints and the impacts between sliders and guides are neglected when the clearance sizes of the translational joints are very small. The contact situations of the normal forces acting on the sliders are described by inequalities and complementarity conditions, while the frictional contacts are characterized by a set-valued force law of the type of Coulomb’s law for dry friction. The dynamic equations of the multibody systems with normal and tangential contact forces are written on the acceleration-force level using the Lagrange multiplier technique. The problem of the transitions of the contact situation of the normal forces acting on sliders and the transitions of the stick-slip of the sliders in the system is formulated as a horizontal linear complementarity problem (HLCP), which is solved by event-driven method. Baumgarte’s stabilization method is used to decrease the constraint drift. Finally, two typical mechanisms are considered as demonstrative application examples. The numerical results obtained show some dynamical behaviors of the systems with frictional translational joints and constraint stabilization effect.     

Hybrid force and motion control of flexible joint parallel manipulators using inverse dynamics approach

An inverse dynamics control algorithm is developed for hybrid motion and contact force trajectory tracking control of flexible joint parallel manipulators. First, an open-tree structure is considered by the disconnection of adequate number of unactuated joints. The loop closure constraint equations are then included. Elimination of the joint reaction forces and the other intermediate variables yield a fourth-order relation between the actuator torques and the end-effector position and contact force variables, showing that the control torques do not have an instantaneous effect on the end-effector contact forces and accelerations because of the flexibility. The proposed control law provides simultaneous and asymptotically stable control of the end-effector contact forces and the motion along the constraint surfaces by utilizing the feedback of positions and velocities of the actuated joints and rotors. A two degree of freedom planar parallel manipulator is considered as an example to illustrate the effectiveness of the method.     

A systematic formulation of dynamic equations for robot manipulators with elastic links

Although a variety of formulation schemes for the dynamic equations of robot manipulators with rigid links can be found in the literature, an efficient method of the formulation for robot manipulators with elastic links is not well known. Accordingly, this work presents the derivation of the equations of motion for application to mechanical manipulators with elastic links. The formulation is conducted analytically using Hamilton's principle. The resultant equations consist of the terms of inertial, Coriolis, centrifugal, gravitational, and exerted forces. They are expressed in terms of a set of independent generalized coordinates. In contrast to conventional variational approaches, the present method provides an efficient and systematic way for obtaining the compact symbolic equations of flexible manipulator systems. Two examples are presented to illustrate the proposed methodology. Firstly, a three-link flexible manipulator with three revolute joints is studied. A flexible manipulator consisting of a prismatic joint and a discrete mass is the second model.     

Robust adaptive controller design for flexible joint manipulators

In this paper the control problem for robot manipulators with flexible joints is considered. A reduced-order flexible joint model is constructed based on a singular perturbation formulation of the manipulator equations of motion. The concept of an integral manifold is utilized to construct the dynamics of a slow subsystem. A fast subsystem is constructed to represent the fast dynamics of the elastic forces at the joints. A composite control scheme is developed based on on-line identification of the manipulator parameters which takes into account the effect of certain unmodeled dynamics and parameter variations. Stability analysis of the resulting closed-loop full-order system is presented. Simulation results for a single link flexible joint manipulator are given to illustrate the applicability of the proposed algorithm.     

Frictional contact analysis of spatial prismatic joints in multibody systems

The contact analysis of spatial prismatic joints remains a hard problem due to its complex nature. In this paper, a methodology for the frictional contact analysis of rigid multibody systems with spatial prismatic joints is presented, which is free of calculating the relative motion between the slider and guide, and is particularly suitable to the case of clearances being tiny. Under the assumption of the slider and guide being rigid, we prove that all types of contacts in the joint can be converted to point-to-point contacts. At each of the candidate points, two gap functions are introduced. However, in the proposed method, not the values of these gap functions but the relations between them are essential. In view of the non-colliding contacts being predominant when clearances of joints are tiny, we formulate the contact forces in terms of resultant frictional forces in the joint, resulting in a linear complementarity problem. By the proposed method, details about the contacts including the impact instants can be obtained, although impacts are not taken into consideration explicitly, as indicated by the numerical examples in this paper.     

A complete strategy for efficient and accurate multibody dynamics of flexible structures with large lap joints considering contact and friction

This paper deals with the dynamics of jointed flexible structures in multibody simulations. Joints are areas where the surfaces of substructures come into contact, for example, screwed or bolted joints. Depending on the spatial distribution of the joint, the overall dynamic behavior can be influenced significantly. Therefore, it is essential to consider the nonlinear contact and friction phenomena over the entire joint. In multibody dynamics, flexible bodies are often treated by the use of reduction methods, such as component mode synthesis (CMS). For jointed flexible structures, it is important to accurately compute the local deformations inside the joint in order to get a realistic representation of the nonlinear contact and friction forces. CMS alone is not suitable for the capture of these local nonlinearities and therefore is extended in this paper with problem-oriented trial vectors. The computation of these trial vectors is based on trial vector derivatives of the CMS reduction base. This paper describes the application of this extended reduction method to general multibody systems, under consideration of the contact and friction forces in the vector of generalized forces and the Jacobian. To ensure accuracy and numerical efficiency, different contact and friction models are investigated and evaluated. The complete strategy is applied to a multibody system containing a multilayered flexible structure. The numerical results confirm that the method leads to accurate results with low computational effort.     

Multibody Dynamics of Very Flexible Damped Systems

An efficient multibody dynamics formulation is presented for simulating the forward dynamics of open and closed loop mechanical systems comprised of rigid and flexible bodies interconnected by revolute, prismatic, free, and fixed joints. Geometrically nonlinear deformation of flexible bodies is included and the formulation does not impose restrictions on the representation of material damping within flexible bodies.The approach is based on Kane's equation without multipliers and the resulting formulation generates 2ndof+m first order ordinary differential equations directly where ndof is the smallest number of system degrees of freedom that can completely describe the system configuration and m is the number of loop closure velocity constraint equations. The equations are integrated numerically in the time domain to propagate the solution.Flexible bodies are discretized using a finite element approach. The mass and stiffness matrices for a six-degree-of-freedom planar beam element are developed including mass coupling terms, rotary inertia, centripetal and Coriolis forces, and geometric stiffening terms.The formulation is implemented in the general purpose multibody dynamics computer program flxdyn. Extensive validation of the formulation and corresponding computer program is accomplished by comparing results with analytically derived equations, alternative approximate solutions, and benchmark problems selected from the literature. The formulation is found to perform well in terms of accuracy and solution efficiency.This article develops the formulation and presents a set of validation problems including a sliding pendulum, seven link mechanism, flexible beam spin-up problem, and flexible slider crank mechanism.     

Reduced-order forward dynamics of multiclosed-loop systems

In this work, a reduced-order forward dynamics of multiclosed-loop systems is proposed by exploiting the associated inherent kinematic constraints at acceleration level. First, a closed-loop system is divided into an equivalent open architecture consisting of several serial and tree-type subsystems by introducing cuts at appropriate joints. The resulting cut joints are replaced by appropriate constraint forces also referred to as Lagrange multipliers. Next, for each subsystem, the governing equations of motion are derived in terms of the Lagrange multipliers, which are based on the Newton–Euler formulation coupled with the concept of Decoupled Natural Orthogonal Complement (DeNOC) matrices, introduced elsewhere. In the proposed forward dynamics formulation, Lagrange multipliers are calculated sequentially at the subsystem level, and later treated as external forces to the resulting serial or tree-type systems of the original closed-loop system, for the recursive computation of joint accelerations. Note that such subsystem-level treatment allows one to use already existing algorithms for serial and tree-type systems. Hence, one can perform the dynamic analyses relatively quickly without rewriting the complete model of the closed-loop system at hand. The proposed methodology is in contrast to the conventional approaches, where the Lagrange multipliers are calculated together at the system level or simultaneously along with the joint accelerations, both of which incur higher order computational complexities, and thereby a greater number of arithmetic operations. Due to the smaller size of matrices involved in evaluating Lagrange multipliers in the proposed methodology, and the recursive computation of the joint accelerations, the overall numerical performances like computational efficiency, etc., are likely to improve. The proposed reduced-order forward dynamics formulation is illustrated with two multiclosed-loop systems, namely, a 7-bar carpet scrapping mechanism and a 3-RRR parallel manipulator.     

A direct integration formulation for exponentially damped structural systems

This paper presents a direct integration formulation for damped structural systems in which damping is characterised by exponential law models. In effect, for non-viscously damped systems the damping forces appearing in the equation of motion are represented by convolution integrals, which take into account the complete history of the load. The proposed formulation employs the Laplace transformation of the motion equation to transform it into a differential equation with time derivative orders higher than two, in contrast to the classical M–C–K–F equation of structural dynamics. Due to the difficulty to manipulate these high order derivatives, the particular case of a damping model with only two kernel functions is taken into account. Then, an implicit direct integration scheme is proposed to discretize the motion equation, which becomes an equivalent second-order M_eq–C_eq–K_eq–F_eq equation. In contrast with other methods revised in the literature, the proposed formulation does not employ internal variables, which normally enlarge the size of the system. Besides, standard direct integration schemes such as those of Newmark’s family can be employed. In the last part of the paper two numerical examples are presented: one for a three degrees-of-freedom system, and another one for a finite element application.     

Dynamic Analysis for Planar Multibody Mechanical Systems with Lubricated Joints

The dynamic analysis of planar multibody systems with revolute clearance joints, including dry contact and lubrication effects is presented here. The clearances are always present in the kinematic joints. They are known to be the sources for impact forces, which ultimately result in wear and tear of the joints. A joint with clearance is included in the multibody system much like a revolute joint. If there is no lubricant in the joint, impacts occur in the system and the corresponding impulsive forces are transmitted throughout the multibody system. These impacts and the eventual continuous contact are described here by a force model that accounts for the geometric and material characteristics of the journal and bearing. In most of the machines and mechanisms, the joints are designed to operate with some lubricant fluid. The high pressures generated in the lubricant fluid act to keep the journal and the bearing surfaces apart. Moreover, the lubricant provides protection against wear and tear. The equations governing the dynamical behavior of the general mechanical systems incorporate the impact force due to the joint clearance without lubricant, as well as the hydrodynamic forces owing to the lubrication effect. A continuous contact model provides the intra-joint impact forces. The friction effects due to the contact in the joints are also represented. In addition, a general methodology for modeling lubricated revolute joints in multibody mechanical systems is also presented. Results for a slider-crank mechanism with a revolute clearance joint between the connecting rod and the slider are presented and used to discuss the assumptions and procedures adopted.     

A constrained motion perspective of railway vehicles collision

Impacts, friction, and normal contact forces occur in the railway vehicles couplers. This paper presents a novel nonsmooth model of the train collision; until now, only penalized models have been approached. The train dynamics is described by an equality of measures formulated at the velocity level. The equations of motion are integrated using the Moreau time-stepping algorithm. Impulsive and normal contact forces are described by a set-valued law of Signorini type, while friction forces are described by a set-valued law of Coulomb type. The constrained forces are computed deducing a particular, simplified formulation of the Udwadia–Kalaba equations. The resulting algorithm is simple and straightforward. Both impulsive and nonimpulsive dynamics are casted in the same framework. Any feature or situation regarding train collisions may be modeled. A demonstrative application is presented. Simulations reveal nonsmooth phenomena like simultaneous multiple collisions, stick-slip, captures, and offset in the final equilibrium position.     

Contact Conditions for Cylindrical,Prismatic, and Screw Joints in Flexible Multibody Systems

This paper focuses on the modeling of the contact conditionsassociated with cylindrical, prismatic, and screw joints in flexiblemultibody systems. In the classical formulation these joints aredeveloped for rigid bodies, and kinematic constraints are enforcedbetween the kinematic variables of the two bodies. These constraintsexpress the conditions for relative translation and rotation of the twobodies along and about a body-fixed axis, and imply the relative slidingand rotation of the two bodies which remain in constant contact witheach other. However, these kinematic constraints no longer implyrelative sliding with contact when one of the bodies is flexible. Toremedy this situation, a sliding joint and a sliding screwjoint are proposed that involves kinematic constraints at theinstantaneous point of contact between the sliding bodies. For slidingscrew joints, additional constraints are added on the relative rotationof the contacting bodies. Various numerical examples are presented thatdemonstrate the dramatically different behavior of cylindrical,prismatic, or screw joints and of the proposed sliding and sliding screwjoints in the presence of elastic bodies, and the usefulness of theseconstraint elements in the modeling of complex mechanical systems.     

Multibody modeling of human body for the inverse dynamics analysis of sagittal plane movements

A multibody methodology for systematic construction of a two-dimensional biomechanical model of a human body is presented, aimed at effective determination of the muscle forces and joint reaction forces in the lower extremities during sagittal plane movements such as vertical jump, standing long jump or jumping down from a height. While the hip, knee and ankle joints are modeled as enforced directly by the muscle forces applied to the foot, shank, thigh and pelvis at the muscle attachment points, the actuation of the other joints is simplified to the torques representing the respective muscle action. The developed formulation is applicable to both the flying and support phases, enhanced by an effective scheme for the determination of reaction forces exclusively in the lower extremity joints. The determination of reactions from the ground is also provided. The problem of muscle force redundancy in the lower extremities is solved by applying the pseudoinverse method, with post-processing procedures used to assure the muscle being tensile. Results of the inverse dynamics analysis of vertical jump are reported.     

Pulse control of flexible multibody systems

An active pulse control method is developed to reduce the vibrations of multibody systems resulting from impact loadings. The pulse, which is a function of system generalized coordinates and velocities, is determined analytically using energy and momentum balance equations of the impacting bodies. Elastic components in the multibody system are discretized using the finite element method. The system equations of motions and nonlinear algebraic constraint equations describing mechanical joints between different components are written in the Lagrangian formulation using a finite set of coupled reference position and local elastic generalized coordinates. A set of independent differential equations are identified by the generalized coordinate partitioning of the constraint Jacobian matrix. These equations are written in the state space formulation and integrated forward in time using a direct numerical integration method. Dependent coordinates are then determined using the constraint kinematic relations. Points in time at which impact occurs are monitored by an impact predictor function, which controls the integration algorithms and forces for the solution of the momentum relation, to define the jump discontinuities in the composite velocity vector as well as the system reaction forces. The effectiveness of the active pulse control in reducing the vibration of flexible multibody aircraft during the touchdown impact is investigated and numerical results are presented.     

Adhesive surface design using topology optimization

Tailoring adhesive properties between surfaces is of great importance for micro-scale systems, ranging from managing stiction in MEMS devices to designing wall-scaling gecko-like robots. A methodology is introduced for designing adhesive interfaces between structures using topology optimization. Structures subjected to external loads that lead to delamination are studied for situations where displacements and deformations are small. Only the effects of adhesive forces acting normal to the surfaces are considered. An interface finite element is presented that couples a penalty contact formulation and a Lennard–Jones model of van der Waals adhesive forces. Two- and three dimensional design optimization problems are presented in which adhesive force distributions are designed such that load-displacement curves of delaminating structures match target responses. The design variables describe the adhesive energy per area of the interface between the surfaces, as well as the geometry of the delaminating structure. A built-in length scale in the formulation of the adhesion forces eliminates the need for filtering to achieve comparable optimal adhesive designs over a range of mesh densities. The resulting design problem is solved by gradient based optimization algorithms evaluating the design sensitivities by the adjoint method. Results show that the delamination response can be effectively manipulated by the method presented. Varying simultaneously both adhesive and geometric parameters yields a wider range of reachable target load-displacement curves than in the case varying adhesive energy alone.     

A nonlinear two-node superelement for use in flexible multibody systems

A nonlinear two-node superelement is proposed for the modeling of flexible complex-shaped links for use in multibody simulations. Assuming that the elastic deformations with respect to a corotational reference frame remain small, substructuring methods may be used to obtain reduced mass and stiffness matrices from a linear finite element model. These matrices are used in the derivation of potential and kinetic energy expressions of the nonlinear two-node superelement. By evaluating Lagrange’s equations, expressions for the internal and external forces acting on the superelement can be obtained. The inertia forces of the superelement are derived in terms of absolute nodal velocities and accelerations, which greatly simplifies the dynamic formulation. Three examples are included. The first two examples are used to validate the method by comparing the results with those obtained from nonlinear beam element solutions. We consider a benchmark simulation of the spin-up motion of a flexible beam with uniform cross-section and a similar simulation in which the beam is simultaneously excited in the out-of-plane direction. Results from both examples show good agreement with simulation results obtained using nonlinear finite beam elements. In a third example, the method is applied to an unbalanced rotating shaft, illustrating the potential of the proposed methodology for a more complex geometry.